From Wikipedia:
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input.

In math, a function has two basic steps:
1) you start with some number, called an "input".
2) the function "DOES STUFF" to the input (squares the number, then adds one, or takes the square root of the input and divides by two, etc.)
3) after you're done, you have the output.
So if the function is f(x) = 3x + 1
a sample "input" would be x = 4
The function "DOES STUFF" to x = 4, specifically, this function takes x , multiplies it by three, and then adds one.
So the OUTPUT is 3*4 +1 = 12 + 1 = 13. :)
Hope that helps.

Well, you can think of relations between people. It's the same thing, only it's more abstract. So a function is a relations. Let's say you have 2 sets of people. Then, you define a relation, like "friendship", and you say that x from the first set is in relation with y from the second set ( x,y are people ) if and only if x is friend with y. You can also say the same thing with the following notation : xRy. For that relation to be a function, x has to be friend with only y. So every person from the first set must have only one person in the second set with which he is in relation. So that's another way of defining functions : a relation for which every x from the domain ( or input ) must have one and only one y in the codomain ( output ) such that xRy. And since this is a function, you can note f(x) = y. Hope this didn't confuse you more :)

Function vs. Relation
Function ==> Each input goes to exactly and only one output. Just as one person cannot be in two places at once, one input cannot go to more than one output.
Relation ==> Each input can go to as many outputs as the function pleases. It's the equivalent of one person being in more than one place at once.